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Theorem issubrg2 20593
Description: Characterize the subrings of a ring by closure properties. (Contributed by Mario Carneiro, 3-Dec-2014.)
Hypotheses
Ref Expression
issubrg2.b 𝐵 = (Base‘𝑅)
issubrg2.o 1 = (1r𝑅)
issubrg2.t · = (.r𝑅)
Assertion
Ref Expression
issubrg2 (𝑅 ∈ Ring → (𝐴 ∈ (SubRing‘𝑅) ↔ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑅,𝑦   𝑥, · ,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦)   1 (𝑥,𝑦)

Proof of Theorem issubrg2
Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 subrgsubg 20578 . . 3 (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
2 issubrg2.o . . . 4 1 = (1r𝑅)
32subrg1cl 20581 . . 3 (𝐴 ∈ (SubRing‘𝑅) → 1𝐴)
4 issubrg2.t . . . . . 6 · = (.r𝑅)
54subrgmcl 20585 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝐴𝑦𝐴) → (𝑥 · 𝑦) ∈ 𝐴)
653expb 1120 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥 · 𝑦) ∈ 𝐴)
76ralrimivva 3201 . . 3 (𝐴 ∈ (SubRing‘𝑅) → ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)
81, 3, 73jca 1128 . 2 (𝐴 ∈ (SubRing‘𝑅) → (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴))
9 simpl 482 . . . 4 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝑅 ∈ Ring)
10 simpr1 1194 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 ∈ (SubGrp‘𝑅))
11 eqid 2736 . . . . . . 7 (𝑅s 𝐴) = (𝑅s 𝐴)
1211subgbas 19149 . . . . . 6 (𝐴 ∈ (SubGrp‘𝑅) → 𝐴 = (Base‘(𝑅s 𝐴)))
1310, 12syl 17 . . . . 5 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 = (Base‘(𝑅s 𝐴)))
14 eqid 2736 . . . . . . 7 (+g𝑅) = (+g𝑅)
1511, 14ressplusg 17335 . . . . . 6 (𝐴 ∈ (SubGrp‘𝑅) → (+g𝑅) = (+g‘(𝑅s 𝐴)))
1610, 15syl 17 . . . . 5 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (+g𝑅) = (+g‘(𝑅s 𝐴)))
1711, 4ressmulr 17352 . . . . . 6 (𝐴 ∈ (SubGrp‘𝑅) → · = (.r‘(𝑅s 𝐴)))
1810, 17syl 17 . . . . 5 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → · = (.r‘(𝑅s 𝐴)))
1911subggrp 19148 . . . . . 6 (𝐴 ∈ (SubGrp‘𝑅) → (𝑅s 𝐴) ∈ Grp)
2010, 19syl 17 . . . . 5 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑅s 𝐴) ∈ Grp)
21 simpr3 1196 . . . . . . 7 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)
22 oveq1 7439 . . . . . . . . 9 (𝑥 = 𝑢 → (𝑥 · 𝑦) = (𝑢 · 𝑦))
2322eleq1d 2825 . . . . . . . 8 (𝑥 = 𝑢 → ((𝑥 · 𝑦) ∈ 𝐴 ↔ (𝑢 · 𝑦) ∈ 𝐴))
24 oveq2 7440 . . . . . . . . 9 (𝑦 = 𝑣 → (𝑢 · 𝑦) = (𝑢 · 𝑣))
2524eleq1d 2825 . . . . . . . 8 (𝑦 = 𝑣 → ((𝑢 · 𝑦) ∈ 𝐴 ↔ (𝑢 · 𝑣) ∈ 𝐴))
2623, 25rspc2v 3632 . . . . . . 7 ((𝑢𝐴𝑣𝐴) → (∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴 → (𝑢 · 𝑣) ∈ 𝐴))
2721, 26syl5com 31 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → ((𝑢𝐴𝑣𝐴) → (𝑢 · 𝑣) ∈ 𝐴))
28273impib 1116 . . . . 5 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢𝐴𝑣𝐴) → (𝑢 · 𝑣) ∈ 𝐴)
29 issubrg2.b . . . . . . . . . . 11 𝐵 = (Base‘𝑅)
3029subgss 19146 . . . . . . . . . 10 (𝐴 ∈ (SubGrp‘𝑅) → 𝐴𝐵)
3110, 30syl 17 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴𝐵)
3231sseld 3981 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑢𝐴𝑢𝐵))
3331sseld 3981 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑣𝐴𝑣𝐵))
3431sseld 3981 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑤𝐴𝑤𝐵))
3532, 33, 343anim123d 1444 . . . . . . 7 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → ((𝑢𝐴𝑣𝐴𝑤𝐴) → (𝑢𝐵𝑣𝐵𝑤𝐵)))
3635imp 406 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢𝐴𝑣𝐴𝑤𝐴)) → (𝑢𝐵𝑣𝐵𝑤𝐵))
3729, 4ringass 20251 . . . . . . 7 ((𝑅 ∈ Ring ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 · 𝑣) · 𝑤) = (𝑢 · (𝑣 · 𝑤)))
3837adantlr 715 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 · 𝑣) · 𝑤) = (𝑢 · (𝑣 · 𝑤)))
3936, 38syldan 591 . . . . 5 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢𝐴𝑣𝐴𝑤𝐴)) → ((𝑢 · 𝑣) · 𝑤) = (𝑢 · (𝑣 · 𝑤)))
4029, 14, 4ringdi 20259 . . . . . . 7 ((𝑅 ∈ Ring ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → (𝑢 · (𝑣(+g𝑅)𝑤)) = ((𝑢 · 𝑣)(+g𝑅)(𝑢 · 𝑤)))
4140adantlr 715 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → (𝑢 · (𝑣(+g𝑅)𝑤)) = ((𝑢 · 𝑣)(+g𝑅)(𝑢 · 𝑤)))
4236, 41syldan 591 . . . . 5 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢𝐴𝑣𝐴𝑤𝐴)) → (𝑢 · (𝑣(+g𝑅)𝑤)) = ((𝑢 · 𝑣)(+g𝑅)(𝑢 · 𝑤)))
4329, 14, 4ringdir 20260 . . . . . . 7 ((𝑅 ∈ Ring ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢(+g𝑅)𝑣) · 𝑤) = ((𝑢 · 𝑤)(+g𝑅)(𝑣 · 𝑤)))
4443adantlr 715 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢(+g𝑅)𝑣) · 𝑤) = ((𝑢 · 𝑤)(+g𝑅)(𝑣 · 𝑤)))
4536, 44syldan 591 . . . . 5 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢𝐴𝑣𝐴𝑤𝐴)) → ((𝑢(+g𝑅)𝑣) · 𝑤) = ((𝑢 · 𝑤)(+g𝑅)(𝑣 · 𝑤)))
46 simpr2 1195 . . . . 5 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 1𝐴)
4732imp 406 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢𝐴) → 𝑢𝐵)
4829, 4, 2ringlidm 20267 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑢𝐵) → ( 1 · 𝑢) = 𝑢)
4948adantlr 715 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢𝐵) → ( 1 · 𝑢) = 𝑢)
5047, 49syldan 591 . . . . 5 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢𝐴) → ( 1 · 𝑢) = 𝑢)
5129, 4, 2ringridm 20268 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑢𝐵) → (𝑢 · 1 ) = 𝑢)
5251adantlr 715 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢𝐵) → (𝑢 · 1 ) = 𝑢)
5347, 52syldan 591 . . . . 5 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢𝐴) → (𝑢 · 1 ) = 𝑢)
5413, 16, 18, 20, 28, 39, 42, 45, 46, 50, 53isringd 20289 . . . 4 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑅s 𝐴) ∈ Ring)
5531, 46jca 511 . . . 4 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝐴𝐵1𝐴))
5629, 2issubrg 20572 . . . 4 (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring) ∧ (𝐴𝐵1𝐴)))
579, 54, 55, 56syl21anbrc 1344 . . 3 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 ∈ (SubRing‘𝑅))
5857ex 412 . 2 (𝑅 ∈ Ring → ((𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴) → 𝐴 ∈ (SubRing‘𝑅)))
598, 58impbid2 226 1 (𝑅 ∈ Ring → (𝐴 ∈ (SubRing‘𝑅) ↔ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  wral 3060  wss 3950  cfv 6560  (class class class)co 7432  Basecbs 17248  s cress 17275  +gcplusg 17298  .rcmulr 17299  Grpcgrp 18952  SubGrpcsubg 19139  1rcur 20179  Ringcrg 20231  SubRingcsubrg 20570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-ress 17276  df-plusg 17311  df-mulr 17312  df-0g 17487  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-grp 18955  df-minusg 18956  df-subg 19142  df-cmn 19801  df-abl 19802  df-mgp 20139  df-rng 20151  df-ur 20180  df-ring 20233  df-subrng 20547  df-subrg 20571
This theorem is referenced by:  opprsubrg  20594  subrgint  20596  issubrg3  20601  issubrgd  21197  cnsubrglem  21435  cnsubrglemOLD  21436  mplsubrg  22026  mplind  22095  dmatsrng  22508  scmatsrng  22527  scmatsrng1  22530  cpmatsrgpmat  22728  elrgspnlem2  33248
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